Coordinate Geometry

Let the coordinates of P be . Since P divides the segment joining and , we use the section formula to find the coordinates of P. The -coordinate of P is 4, so the ratio of division is: Thus, the ratio of division is 1:1, and P divides the line segment equally. The -coordinate of P is the average of the -coordinates of the endpoints: Thus, the correct -coordinate is .

We are given the coordinates of points A and B as: The total length of the wire is the distance between points A and B, and the points where the lanterns are hung divide the wire into three equal parts. Therefore, we need to find the points that trisect the wire. Let the coordinates of the points that trisect the wire be denoted as and , such that: and ### 1. Finding the Coordinates of : To find the coordinates of , we use the section formula. The section formula states that the coordinates of a point dividing a line segment in the ratio are given by: Here, we are dividing the line segment in the ratio , so and . Applying the section formula: So, the coordinates of are: ### 2. Finding the Coordinates of : Similarly, to find the coordinates of , we divide the line segment in the ratio . Using the section formula with and : So, the coordinates of are: Thus, the coordinates of the points where the lanterns are hung are and .

The given equation of the line is: This represents a parametric form of the line. We can express in terms of the parameter : Let the coordinates of point be . To find the distance between and , we use the distance formula: Substitute the values of from the parametric equations: Simplify the equation and solve for . After solving, we find: Substitute back into the parametric equations to find the coordinates of : Thus, the coordinates of are .

We are given the coordinates of points A and B as: The total length of the wire is the distance between points A and B, and the points where the lanterns are hung divide the wire into three equal parts. Therefore, we need to find the points that trisect the wire. Let the coordinates of the points that trisect the wire be denoted as and , such that: and 1. Finding the Coordinates of : To find the coordinates of , we use the section formula. The section formula states that the coordinates of a point dividing a line segment in the ratio are given by: Here, we are dividing the line segment in the ratio , so and . Applying the section formula: So, the coordinates of are: 2. Finding the Coordinates of : Similarly, to find the coordinates of , we divide the line segment in the ratio . Using the section formula with and : So, the coordinates of are: Thus, the coordinates of the points where the lanterns are hung are and .